Mathematical Analysis (2nd International Edition) by Tom Apostol

By Tom Apostol

It offers a transition from effortless calculus to complex classes in actual and complicated functionality thought and introduces the reader to a couple of the summary pondering that pervades smooth research.

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The ball B(a; r) consists of all points whose distance from a is less than r. In R1 this is simply an open interval with center at a. In R2 it is a circular disk, and in R3 it is a spherical solid with center at a and radius r. 5 Definition of an interior point. Let S be a subset of R", and assume that a e S. Then a is called an interior point of S if there is an open n-ball with center at a, all of whose points belong to S. In other words, every interior point a of S can be surrounded by an n-ball B(a)s S.

Now define h by the equation h(x) = 1 + gx(x) if x e R, and show that h f S. 18 Let S be the collection of all sequences whose terms are the integers 0 and 1. Show that S is uncountable. 19 Show that the following sets are countable: a) the set of circles in the complex plane having rational radii and centers with rational coordinates, b) any collection of disjoint intervals of positive length. 20 Let f be a real-valued function defined for every x in the interval 0 < x < 1. Suppose there is a positive number M having the following property: for every choice of a finite number of points x1, x2, ...

However, since much of the theory of sets does not depend on the nature of the individual objects in the collection, we gain a great economy of thought by discussing sets whose elements may be objects of any kind. It is because of this quality of generality that the theory of sets has had such a strong effect in furthering the development of mathematics. 2 NOTATIONS Sets will usually be denoted by capital letters : A, B, C, ... , X, Y, Z, and elements by lower-case letters: a, b, c, ... , x, y, z.

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